For closed-loop data, the REDAR algorithm (VARX fit plus balanced reduction) has one-step-ahead prediction error bounded by the optimal error plus terms that decay with model order p and with sample size T as O(1/√T).
Nonasymptotic bounds for autore- gressive time series modeling,
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Non-asymptotic Closed-Loop System Identification using Autoregressive Processes and Hankel Model Reduction
For closed-loop data, the REDAR algorithm (VARX fit plus balanced reduction) has one-step-ahead prediction error bounded by the optimal error plus terms that decay with model order p and with sample size T as O(1/√T).